10.14 Finding Taylor or Maclaurin Series for a Function

Building the Taylor Series

A Taylor series for f(x) centered at x = a is built from derivatives: f(a) + f'(a)(x−a) + f''(a)/2!·(x−a)² + f'''(a)/3!·(x−a)³ + ... The general term is f^(n)(a)/n! · (x−a)^n. A Maclaurin series is just a Taylor series centered at a = 0.

Using Known Series

Instead of computing derivatives repeatedly, you can manipulate known Maclaurin series (for e^x, sin x, cos x, 1/(1−x), ln(1+x)) via substitution, multiplication by x^n, differentiation, or integration to quickly find new series. This is faster and less error-prone than direct differentiation for many functions.

Common Maclaurin Series to Memorize

e^x = Σ x^n/n!, sin x = Σ (−1)^n x^(2n+1)/(2n+1)!, cos x = Σ (−1)^n x^(2n)/(2n)!, 1/(1−x) = Σ x^n (|x|<1). Memorizing these lets you build new series for related functions like e^(−x²) or x·cos(x³) almost instantly.

Key terms

Taylor series formula

f(x) = Σ f^(n)(a)/n! · (x−a)^n, built from derivatives of f evaluated at center a.

Maclaurin series

A Taylor series centered at a = 0.

Maclaurin series for e^x

Σ x^n/n! = 1 + x + x²/2! + x³/3! + ...

Maclaurin series for sin x

Σ (−1)^n x^(2n+1)/(2n+1)! = x − x³/3! + x⁵/5! − ...

Trick for new series from known ones

Use substitution, multiplication by x^n, or term-by-term differentiation/integration on known series (e^x, sin x, cos x, 1/(1−x)) instead of computing derivatives directly.

Try an AP-style question

Which is the Maclaurin series for eˣ?

1. Σ (−1)ⁿ xⁿ/n!
2. Σ xⁿ/n!
3. Σ xⁿ
4. Σ xⁿ/n